Let f(x)=sin^-1(5x+1). Find f'(x).

Answer 1

Recall that sin^-1(x) is arcsin(x) which is defined only for -1 <= x <=1

and that d arcsin(x)/dx = 1/√(1-x²).

So, you are looking for d/dx arcsin(5x+1) and your function is a composite function and you need to use

the chain rule.. Hence:

d arcsin(5x+1) /dx = d arcsin(5x+1)/ d(5x+1) * d(5x+1)/dx

I trust that you can continue from here. Also state that for your function to be defined you need to have

-1 <= 5x+ 1 <= 1 which implies - 2/5 <= x <= 0

Answer 2

Sin(y) = 5x+1

Cos(y)y' = 5

y' = 5/cos(y) or 5sec(y). Now use a right triangle to figure out what cos(y) (or sec(y)) equals in terms of x using the triangle you get from knowing that sin(y) = (5x+1)/1